Lecture 14

**Scribe: Deepak Ajwani**

**20th September, 2002**

In the last lecture, we discussed *Bargain Solution* for
the two-player case where the utilities are linear. The game
terminates in disagreement after N rounds and both the player get a
zero payoff. In case the utilities are linear, the set of feasible
outcomes (also called bargaining set) looks as shown below.

If the utilities of the players are consistent with Van Neumann Morganstern utility postulates, then the bargaining set will always be convex. To see this, consider two outcomes and , with utilities given by and . Now consider an outcome : with probability and with prob . The utility for player 1 on w(p) is . Similarly, . So, is a point in the line joining and , and so, is in the bargaining set. The above analysis implies that any convex combination of two outcomes in the bargaining set is also in the bargaining set. Therefore, we will assume that the Bargaining set is a convex region.

Let us recall the definition of Subgame Perfect Equilibrium. A strategy profile is a Subgame Perfect Equilibrium of a game if it is a Nash equilibrium of every subgame of the game.

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() | () If player 1 will offer to player 2, the player 2 being rational won't reject it, as she knows that if she rejects she will get (), which is less than (). And since, for player 1, () (), he will have no hesitation in offering this to player 2. This will result in termination of decision tree at the first level itself, which contradicts the assumption taken above that subgame perfect equilibrium was attained after the rejection of first proposal of player 1 by player 2. Hence, it is proved that any subgame perfect equilibrium can be achieved in the first turn itself. |

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Deepak Ajwani 2002-11-22